Fundamental Theorem of Calculus (FTC)

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Bloom’s Taxonomy for different levels of understanding
Figure 1.1: Bloom's Taxonomy

 

FTC-II

Theorem: Fundamental Theorem of Calculus - Part II [FTC-II]

Let f(t) be continuous on [a,b]. Let F(t) be any antiderivative of f(t). Then abf(t)dt=F(b)F(a).

Check Conditions of FTC-II

  1. f(t) is the black graph. Is f(t) continuous on [1,12]? Explain. NOTE: A function is continuous on an interval IF:

    • there are NO holes in the graph;

    • there are NO jump discontinuities in the graph; and

    • there are NO infinite discontinuities in the graph.

Activity 16

NOTE: This DESMOS activity will help you understand the Fundamental Theorem of Calculus - Part II (FTC-II). For a good proof (which is beyond the expectation for this course) of both FTC-I and FTC-II, see https://math.berkeley.edu/~peyam/Math1AFa10/Proof of the FTC.pdf.

Procedure:

Let g(x)=1af(t)dt. In the DESMOS page,

  1. Move the slider for the parameter a [Box 3] to a value of a=12.

  2. Identify the net area [the value listed in Box 2] under f(t) [black curve].

  3. From the graph, identify the value of the antiderivative F(x) function [green curve] when a=12.

  4. From the graph, identify the value of the antiderivative F(x) function [green curve] when a=1.

  5. Evaluate F(12)F(1) and compare to the net area under f(t) [black curve]. What do you notice?

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Created: Tuesday, 25 August 2020 03:53 EDT Last Modified: Monday, 30 May 2022 - 14:32 (EDT)